The Bounded-Linear Dichotomy Holds for all 4 Point Posets

Abstract: Given a finite poset $\mathcal P$, its induced saturation number, $\text{sat}^*(n,\mathcal P)$ is the smallest size a family of subsets of $[n]$ can have such that it does not contain an induced copy of $\mathcal P$, but adding any other set to it creates one. The saturation numbers already show a sharp dichotomy -- Freschi, Piga, Sharifzadeh and Treglown showed that for any given poset its saturation number is either bounded or at least $2\sqrt n$. The dominant conjecture is that in fact, the saturation numbers are either bounded, or exactly linear. In this paper we show that this dichotomy is true for all posets on at most 4 points. Most of these posets, most notably the butterfly and the diamond, had individually long resisted analysis, and the missing piece which this paper tackles is the poset comprised of a $\mathcal V$ and an isolated point, denoted by $\widehat{\mathcal V}$, for which we show linear saturation. The architecture of the proof uncovers the structure of an arbitrary $\widehat{\mathcal V}$- saturated family, namely an antichain with at most two chains above it, and an upside-down forest below it. Moreover, we anticipate that, at least partially, this structure is relevant to the wider class of posets that contain an isolated point.